Almost regular cages

Abstract

Let k be an integer greater than 2. We consider two classes of graphs, which are near to being k-regular. We say that a simple graph is almost regular of Type I if all of its vertices, but one, have degree k, and the exceptional vertex has degree 2. The graph is almost regular of Type II if all of its vertices, but two, have degree k, and the exceptional vertices have degree k − 1. We show that there is a close relationship between the problem of finding small almost regular graphs of girth g and that of finding small (k,g)-graphs (the cage problem).